Zero‐Sum Risk‐Sensitive Stochastic Differential Games

We study zero‐sum risk‐sensitive stochastic differential games on the infinite horizon with discounted and ergodic payoff criteria. Under certain assumptions, we establish the existence of values and saddle‐point equilibria. We obtain our results by studying the corresponding Hamilton–Jacobi–Isaacs equations. Finally, we show that the value of the ergodic payoff criterion is a constant multiple of the maximal eigenvalue of the generators of the associated nonlinear semigroups.